The Edgeworth series or Gram-Charlier A series, named in honor of Francis Ysidro Edgeworth, are series that approximate a probability distribution in terms of its cumulants. Edgeworth Francis Ysidro Edgeworth (February 8, 1845 - February 13, 1926) was an Irish polymath who studied at Trinity College, Dublin before obtaining a scholarship to Balliol College, Oxford where he subsequently became a professor. ... In mathematics, a series is the sum of a sequence of terms. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... // Cumulants of probability distributions In probability theory and statistics, the cumulants κn of the probability distribution of a random variable X are given by In other words, κn/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function. ...

Gram-Charlier A series

The key idea of these expansions is to write the characteristic function of the distribution whose probability density function is F to be approximated in terms of the characteristic function of a distribution with known and suitable properties, and to recover F through the inverse Fourier transform. Some mathematicians use the phrase characteristic function synonymously with indicator function. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...

Let f be the characteristic function of the distribution whose density function is F, and κ_r its cumulants. We expand in terms of a known distribution with probability density function Ψ, characteristic function ψ, and cumulants γ_r. The density Ψ is generally chosen to be that of the normal distribution, but other choices are possible as well. By the definition of the cumulants, we have the following formal identity: The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields, especially in physics and engineering. ...

By the properties of the Fourier transform, (it)^rψ(t) is the Fourier transform of (−1)^r D^r Ψ(x), where D is the differential operator with respect to x. Thus, we find for F the formal expansion

If Ψ is chosen as the normal density with mean and variance as given by F, that is, mean μ = κ₁ and variance σ² = κ₂, then the expansion becomes

By expanding the exponential and collecting terms according to the order of the derivatives, we arrive at the Gram-Charlier A series. If we include only the first two correction terms to the normal distribution, we obtain

with h₃ = (x³ − 3x)/3! and h₄ = (x⁴ − 6x² + 3)/4! (these are Hermite polynomials). Note that this expression is not guaranteed to be positive, and is therefore not a valid probability distribution! The Gram-Charlier A series diverges in many cases of interest. [The series converges only if f(x) falls off faster than exp(−x²/4) at infinity (Cramér 1957).] When it does not converge, the series is also not a true asymptotic expansion, because it is not possible to estimate the error of the expansion. Therefore, the Edgeworth series (see next section) is generally preferred over the Gram-Charlier A series. In mathematics, the Hermite polynomials, named in honor of Charles Hermite (pronounced air MEET), are a polynomial sequence defined either by (the probabilists Hermite polynomials), or sometimes by (the physicists Hermite polynomials). These two definitions are not exactly equivalent; either is a trivial rescaling of the other. ... In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ...

Edgeworth series

Edgeworth developed a similar expansion as an improvement to the central limit theorem. The advantage of the Edgeworth series is that the error is controlled, so that it is a true asymptotic expansion. Central limit theorem - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ...

Let X_i be a sequence of identically distributed random variables, and Y_n the standardized sum A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...

Further, let F_n be the probability density function of the variables Y_n. By the central limit theorem,

for every x, as long as the means and variances are finite and the sum of variances diverges to infinity. (Generally, the conclusion of the central limit theorem is about the limit of cumulative distribution functions, not of probability density functions, and therefore applies to discrete distributions as well. But discrete distributions are not contemplated in the present context). In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or...

Now assume that the random variables X_i have mean μ, variance σ², and higher cumulants κ_r=σ^rλ_r. If we expand in terms of the unit normal distribution, that is, if we set

then the cumulant differences in the formal expression of the characteristic function f_n(t) of F_n are

The Edgeworth series is developed similarly to the Gram-Charlier A series, only that now terms are collected according to powers of n. Thus, we have

where P_j(x) is a polynomial of degree 3j. Again, after inverse Fourier transform, the density function F_n follows as In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

The first three terms of the expansion are (Cramér 1957)

Here, Ψ^(j)(x) is the jth derivative of Ψ(x) with respect to x. Blinnikov and Moessner (1998) have given a simple algorithm to calculate higher-order terms of the expansion.

Further reading

• S. Blinnikov and R. Moessner (1998). "Expansions for nearly Gaussian distributions". Astron. Astrophys. Suppl. Ser. 130:193-205.
• Harald Cramér (1957). Mathematical Methods of Statistics. Princeton University Press, Princeton.
• D. L. Wallace (1958). "Asymptotic approximations to distributions". Ann. Math. Stat. 29:635-654.